Lecture Notes in Computer Science 3472

(12.4)
✟
nat(X ′ )
{A/0, B/X ′ , X /X ′ }
12 Technology of Test-Case Generation 335
max(A, B, X )
✟ ❍
✟
✟ ❍❍❍❍
✟
✟
(12.6)
(12.5)
❍
nat(X ′ )
{A/s(X ′ ), B/0, X /s(X ′ )}
max(A ′ , B ′ , X ′ )
{A/s(A ′ ), B/s(B ′ ), X /s(X ′ )}
Fig. 12.2. Resolution tree of max(A, B, X )
It is easy to see that every equation of the form f (t1,...,tn) =t can be
replaced by a literal ˜ f (t1,...,tn, t) witharityn + 1. Hence, we obtain a Horn
clause resp. Prolog rule of the form
˜f (t1,...,tn, r) ← ˜ f1(t1,1,...,t1,n1, r1),..., ˜ fm(tm,1,...,tm,nm, rm).
Therefore, we have transformed the initial specification into a Prolog program.
It remains to show how test cases can be generated with this program.
Two important properties in this context are completeness and termination. To
ensure a complete proof search strategy, iterative deepening can be used instead
of Prolog’s standard depth-first search. This, however, does not guarantee
termination because some unsatisfiable goals cannot be detected. To solve this
problem, rewriting is used by Bernot et al. [BGM91] to simplify goals before
each resolution step in order to alleviate the detection of unsatisfiable goals.
The refinement resp. unfolding process to obtain a suitable decomposition of
the specification into sub-domains is implemented by recursively replacing each
defined operator by the cases corresponding to its definition. This is already
provided by Prolog’s resolution principle. For instance, consider the predicate
max/3 defined by the following three clauses, where s denotes the successor
function:
max(0, X , X ) ← nat(X ). (12.4)
max(s(X ), 0, s(X )) ← nat(X ). (12.5)
max(s(A), s(B), s(X )) ← max(A, B, X ). (12.6)
For example, max(s(0), s(s(s(0))), s(s(s(0)))) is true since the maximum of
s(0) . =1ands(s(s(0))) . =3iss(s(s(0))) . = 3. The resolution tree shown in
Fig. 12.2 is obtained when applying one resolution step to the goal max(A, B, X ).
Theleavesofthistreeobviouslyrepresent a decomposition of max/3 according
to its definition. For the first two clauses we obtain the resolvent nat(X ′ ), and
for the third clause we obtain the resolvent max(A ′ , B ′ , X ′ ) together with the
corresponding unifiers. These resolvents can now be further decomposed. The
crucial point is to decide when the refinement process has to be stopped, i.e., to
control the degree of decomposition. To this aim, meta-clauses are used by Bernot
et al. [BGM91]. The literals are chosen for resolution according to a selection